My thanks to several people who made suggestions in answer to my question
about the history of 20C mathematics in its relation to computing. After
checking out most of the sources recommended and straying into others, I'd
like to draw your attention to the following, in case my quest is of more
general interest:

Of all of the books I examined, this is the closest to what I was hoping to
find: clearly written, non-technical and quite insightful at key points. It
emphasizes one of the histories of computing, as the title suggests, namely
the mathematical-logical, and more or less ends the story with Turing's own
end. The story it tells hangs together by a combination of biographical and
intellectual themes, but these are very well balanced. Readings from it
would suit advanced undergraduate and (post)graduate students in the
humanities quite well.

(Anecdotal aside. The basic historiographical problem with any account like
this emerged in a Freudian typo a few minutes ago. I was recommending the
book to a young, very bright and eager nephew; I typed, The Universal
Computer: The Road from Turing to Leibniz!.... I am not implying a
criticism of Davis's fine account, rather the point that it is only one of
the histories, the one you get when you take "the" computer to be in
essence defined by what Turing did.)

2. David Hilbert, "Axiomatic Thought" (1918), in William Ewald, ed., From
Kant to Hilbert: A Source Book in the Foundations of Mathematics. Volume
II. Oxford: Oxford University Press, 1996. This paper is entirely
non-technical and a very fine example of how Hilbert thought about
mathematical thinking. It provides an illuminating definition of the
meaning of the word "theory" in mathematics and best of all a very clear
statement of his axiomatic method. The editor's prefatory comments on the
mistaken notion that Hilbert was simply a "formalist" are quite helpful.
One of course needs reference to other papers by Hilbert, including the
famous 1900 address, which is online.

3. Kurt Gödel, "The Modern development of the foundations of mathematics in
the light of philosophy", denoted as Gödel 1961/?, in Unpublished essays
and lectures, vol. III of the Collected Works, ed. Solomon Feferman et al
(Oxford: Oxford University Press, 2001), with a very helpful preface by
Dagfinn Fĝllesdal (pp. 364-72); the essay itself in English is online at
http://www.marxists.org/reference/subject/philosophy/works/at/godel.htm.
This, it seems, was an address that he wrote but never delivered. In it he
talks about Hilbert's Program, how he thinks it failed and he charts what
he regards as the most promising "middle way" forward -- which is through
Husserlian phenomenology to deepen the abstract concepts on which
mechanical, formalist schemes are founded. The project to deepen
foundations he shares, of course, with Hilbert and many others; what is
especially valuable here is the pointer into phenomenology (on which
Fĝllesdal comments at length). The result of the deepening is brought out
by Davis, p. 124: "For any specific given formalism there are mathematical
questions that will transcend it. On the other hand, in principle, each
such question leads to a more powerful system which enables the resolution
of that question. One envisions hierarchies of ever more powerful systems
each making it possible to decide questions left undecidable by weaker
systems." This is immediately recognizable as the perfective cycle of
modelling. But I am left full of questions about the phenomenology!

There are a number of other, I suppose obvious, items, such as Turing's
1936 paper and von Neumann's First Draft of a Report on the EDVAC (1945),
although these are well covered in Davis' book, which does an excellent job
of connecting Turing's work to von Neumann's.

But lest this somehow seem a claim of first discovery within our small
circle, allow me to note that Tito Orlandi has been insisting on the
importance of the mathematical topics for many years.